ISSN 2348-1196 (print) International Journal of Computer Science and Information Technology Research ISSN 2348-120X (online) Vol. 8, Issue 4, pp: (46-49), Month: October - December 2020, Available at: www.researchpublish.com
Fractional Clairaut’s Differential Equation and Its Application Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong Province, China
Abstract: This paper uses a new multiplication of fractional functions and the product rule and chain rule for fractional derivatives, regarding the Jumarie type of modified Riemann-Liouville (R-L) fractional derivative, to obtain the general solution and singular solution of fractional Clairaut’s differential equation. On the other hand, an example is proposed to illustrate our results. Keyword: New multiplication, Product rule, Chain rule, Modified R-L fractional derivative, Fractional Clairaut’s differential equation.
I. INTRODUCTION Fractional calculus belongs to the field of mathematical analysis which involves the investigation and applications of integrals and derivatives of arbitrary order. Although fractional calculus has almost the same long history as the classical calculus, it was only in recent decades that its theory and applications have rapidly developed. Oldham and Spanier [1] published the first monograph in 1974. Ross [2] edited the first proceedings that was published in 1975. Thereafter theory and applications of fractional calculus have attracted much interest and have become a vibrant research area. Nowadays, the number of monographs and proceedings devoted to fractional calculus is already large, e.g. [3-8]. Fractional differential equations arise in many complex systems in nature and society with many dynamics, such as charge transport in amorphous semiconductors, the spread of contaminants in underground water, relaxation in viscoelastic materials like polymers, the diffusion of pollution in the atmosphere, and many more [9-10]. However, the problem of studying fractional differential equations has been dealt with by numerous authors throughout history, particularly in recent years [11-12]. A wide description of the existence and uniqueness of solutions of initial value problem for fractional order differential equations together with its applications can be found in the literature [13]. Other papers on fractional differential equations can refer to [15-21]. Unlike standard calculus, there is no unique definition of derivation and integration in fractional calculus. The commonly used definition is the Riemann-Liouville (R-L) fractional derivative. Other useful definitions include Caputo definition of fractional derivative, the Grunwald-Letinikov (G-L) fractional derivative, and Jumarie’s modified R-L fractional derivative is used to avoid nonzero fractional derivative of constant functions. In this paper, the general solution and singular solution of fractional Clairaut’s differential equation can be obtained by using a new multiplication of fractional functions, and product rule and chain rule for fractional derivatives, regarding the Jumarie type of modified R-L fractional derivative. Moreover, the singular solution of fractional Clairaut’s differential equation is the fractional envelope of the general solution curve family. In fact, the result we obtained is the generalization of Clairaut’s ordinary differential equation. On the other hand, we provide an example to demonstrate the application of our results.
II. PRELIMINARIES AND RESULTS Firstly, we introduce the fractional calculus used in this article. Definition 2.1: Let be a real number and Jumarie type is defined by ([14])
be a positive integer Then the modified R-L fractional derivatives of
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